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Mathematics
Linear and Abstract Algebra
Algebraic Extensions - Dummit and Foote, Propn 11 and 12 ....
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[QUOTE="Math Amateur, post: 5763198, member: 203675"] I am reading Dummit and Foote, Chapter 13 - Field Theory. I am currently studying Section 13.2 : Algebraic Extensions I need some help with an aspect of Propositions 11 and 12 ... ... Propositions 11 and 12 read as follows:[ATTACH=full]203616[/ATTACH] [ATTACH=full]203617[/ATTACH] Now Proposition 11 states that the degree of ##F( \alpha )## over ##F## is [U][I][B]equal to[/B][/I][/U] the degree of the minimum polynomial ... ... that is ##[ F( \alpha ) \ : \ F ] = \text{ deg } m_\alpha (x) = \text{ deg } \alpha##... ... BUT ... ...... ... Proposition 12 states that ... "if ##\alpha## is an element of an extension of degree ##n## over ##F##, then ##\alpha## satisfies a polynomial of degree [U][I][B]at most ##n## over ##F##[/B][/I][/U] ... ... "Doesn't Proposition 11 guarantee that the polynomial (the minimum polynomial) is actually of degree equal to ##n##?Can someone please explain in simple terms how these statements are consistent?Help will be appreciated ... Peter [/QUOTE]
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Algebraic Extensions - Dummit and Foote, Propn 11 and 12 ....
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